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Introduction to computational algebraic statistics. (English) Zbl 1442.62765
Iohara, Kenji (ed.) et al., Two algebraic byways from differential equations: Gröbner bases and quivers. Cham: Springer. Algorithms Comput. Math. 28, 185-212 (2020).
Summary: In this paper, we introduce the fundamental notion of a Markov basis, which is one of the first connections between commutative algebra and statistics. The notion of a Markov basis is first introduced by P. Diaconis and B. Sturmfels [Ann. Stat. 26, No. 1, 363–397 (1998; Zbl 0952.62088)] for conditional testing problems on contingency tables by Markov chain Monte Carlo methods.
For the entire collection see [Zbl 1444.14002].
62R01 Algebraic statistics
62H17 Contingency tables
62-08 Computational methods for problems pertaining to statistics
65C05 Monte Carlo methods
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
Full Text: DOI
[1] 4ti2 team, 4ti2—a software package for algebraic, geometric and combinatorial problems on linear spaces.
[2] A. Agresti. A survey of exact inference for contingency tables. Statist. Sci. 7(1), 131-177, (1992). With comments and a rejoinder by the author. · Zbl 0955.62587
[3] A. Agresti. in Categorical Data Analysis. Wiley Series in Probability and Statistics, 2nd edn. (Wiley-Interscience [Wiley], Hoboken, NJ, 2013).
[4] S. Aoki, H. Hara, A. Takemura, Markov Bases in Algebraic Statistics, Springer Series in Statistics (Springer, New York, 2012). · Zbl 1304.62015
[5] J. Cornfield, A statistical problem arising from retrospective studies, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. IV (University of California Press, Berkeley and Los Angeles, 1956), pp. 135-148.
[6] D. Cox, J. Little, D. O’Shea, in Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics, 3rd edn. (Springer, New York, 2007). An introduction to computational algebraic geometry and commutative algebra.
[7] W. Decker, G.M. Greuel, G. Pfister, H. Schönemann, in Singular 3-1-2, a Computer Algebra System for Polynomial Computations. · Zbl 1344.13002
[8] P. Diaconis, B. Sturmfels, Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26(1), 363-397 (1998). · Zbl 0952.62088
[9] M. Gail and N. Mantel. Counting the number of \(r\times c\) contingency tables with fixed margins. J. Am. Statist. Assoc. 72(360, part 1), 859-862 (1977). · Zbl 0372.62042
[10] D.R. Grayson and M.E. Stillman, Macaulay2, A Software System for Research in Algebraic Geometry.
[11] S.J. Haberman, A warning on the use of chi-squared statistics with frequency tables with small expected cell counts. J. Am. Statist. Assoc. 83(402), 555-560 (1988). · Zbl 0648.62047
[12] W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1), 97-109 (1970). · Zbl 0219.65008
[13] T. Hibi (ed.), in Gröbner bases, Japanese edn. (Springer, Tokyo, 2013). Statistics and software systems.
[14] E.L. Lehmann, J.P. Romano, Testing Statistical Hypotheses, 3rd edn., Springer Texts in Statistics (Springer, New York, 2005). · Zbl 1076.62018
[15] N. Noro, N. Takayama, H. Nakayama, K. Nishiyama, K. Ohara, Risa/asir. A Computer Algebra System.
[16] L. Pachter, B. Sturmfels (eds.), Algebraic Statistics for Computational Biology (Cambridge University Press, New York, 2005). · Zbl 1108.62118
[17] R.L. Plackett, in The Analysis of Categorical Data, Griffin’s Statistical Monograph Series, vol. 35, 2nd edn. (Macmillan Co., New York, 1981). · Zbl 0479.62046
[18] C. Team, inA System for Doing Computations in Commutative Algebra.
[19] A.
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